Female fish and shellfish produce astounding numbers of eggs, giving any population the capacity to increase its density rapidly after a perturbation if conditions are right for the survival of the young. This rapid reproductive rate (r-selected species) allows humans to harvest fish populations and anticipate their recovery. The degree to which a stock may be harvested has historically been determined by the form of the S/R relationship.

The S/R relationship is normally presented graphically as a scatter plot with the number of females in the spawning stock on the abscissa (x-axis) and the number of recruits on the ordinate (y-axis). The spawning stock is defined, normally, as the number of female organisms in the population of reproductive age and able to reproduce in any one year. The recruits are defined as those young who survive to either maturity, or to be captured by the fishery. Each point in this graph will represent two values observed during systematic sampling during different years. As an example, consider a species in which young recruit into the fishery at the age of 2. Thus, the spawning stock in 1998 produced the recruits observed in 2000. If there were 4 million adult females in 1998, and three million recruits in 2000, then a point would be placed at (4 000 000, 3 000 000) in our S/R graph, representing the cohort for the year 2000. In order to characterize this relationship statistically, which is necessary for all management models, many years of data must be collected on each stock of interest.

The S/R relationship is normally dome-shaped, facing down. This means that we expect zero or very few recruits when the spawning stock is very low (in other words, the relationship passes through the origin), that we have maximal recruitment for a middling number of spawners, and that recruitment is badly reduced if there are too many mature adults. This latter point is best understood if we realize that adult and immature fish often compete for food, with the larger adults winning this competition. Thus, if there are many adults, survival rates of the young and immature fish will be very low, leading to low recruitment rates.

There are two classical mathematical models used to describe the relationship between the stock and the number of recruits. The first is called the Beverton-Holt model, which states that R=E/(E+g*Rmax)*Rmax, where g is a parameter, R is the number of recruits, E is the egg production (number of females * average egg production). Shortly thereafter, Ricker suggested the following model (now called the Ricker curve): R=R1*E-R2*E where R1 and R2 are parameters. More recently, Deriso and Schnute have proposed a more general model, which reduces to either of the former models when certain parameters attain some value. Their model is: R=R1*E*(1-R2*R3*3)1/R3.

These models, and some variants, have been used to manage fish stocks for the past fifty years. In recent years they have come under criticism for a number of reasons, both theoretical and practical. On the theoretical side, they do not account for systematically changing environmental conditions, changes in the water currents or immigration/emigration. The practical problems are that, despite a good theoretical foundation, they have a remarkably poor track record. Many enormous fish stocks have been carefully managed into near-extinction by the use of these models (eg. Atlantic cod, the anchovy, the salmon). Modern management approaches still consider the S/R relationship when formulating their harvesting recommendations, but they are only one of many different approaches used in an integrative manner.